1.

Introduction: In supergravity [1]}, [2]}, [3]}, [4]}, [5]}, [6]} we start out with Einstein's theory, but it needs to be formulated with a vierbein \( e_{\mu a} \)  and a spin connection
\(w_{\mu ab}\) , instead of the metric tensor \(g_{\mu \nu }\)  and an affine connection \(\Gamma _{\mu \nu \lambda }\) . The vierbein and spin connection are needed so that the gravitino \( \Psi _{\mu \alpha }\)  can be coupled to gravity. All three of these fields are gauge fields, and their gauge transformations are of the form
\(\delta e_{\mu }^{\;\; a} = \partial _{\mu } \xi ^a + \cdots ;\;\delta w_{\mu }^{\;\; a b} = \partial _{\mu } \rho ^{ab} + \cdots ;\; \delta \Psi _{\mu \alpha } = \partial _{\mu } C_{\alpha } + \cdots \) 
